View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Dagstuhl Research Online Publication Server Proof Presentation Jörg Siekmann DFKI Saarbrücken, DE [email protected] The talk is based on a forthcoming book is about the human-oriented pre- sentation of a mathematical proof in naturallanguage, in a style as we may nd it in a typical mathematical text book. How can a proof be other than human-oriented? What we have in mind is a deduction systems, which is implemented on a computer, that proveswith some human interactiona mathematical textbook as may be used in an undergrad- uate course. The proofs generated by these systems today are far from being human-oriented and can in general only be read by an expert in the respective eld: proofs between several hundred (for a common mathematical theorem), for more than a thousand steps (for an unusually dicult theorem) and more than ten thousand deduction steps (in a program verication task) are not uncom- mon. 1 Although these proofs are provably correct,they are typically marred by many problems: to start with, that are usually written in a highly specialised logic such as the resolution calculus, in a matrix format, or even worse, they may be generated by a model checker. Moreover they record every logical step that may be necessary for the minute detail of some term transformation (such as, for example, the rearrangement of brackets) along side those arguments, a mathematician would call important steps or heurekasteps that capture the main idea of the proof. Only these would he be willing to communicate to his fellow mathematiciansprovided they have a similar academic background and work in the same mathematical discipline. If not, i.e. if the proof was written say for an undergraduate textbook, the option of an important step may be viewed dierently depending on the intended reader. Now, even if we were able to isolate the ten important steps out of those hundreds of machine generated proof steps there would still be the startling 1 The Argonne deduction system OTTER, the old MKRP-System or more recently the SPASS system from Saarbrücken, SETHEO in München or Voronkov's System VAMPIRE can generate and represent search spaces of more than a billion clauses and the proof as thus found may be up to several hundred or even more than a thousand steps long. The interactive VSE program verication system of the DFKI at Saarbücken has generated proofs of more than ten thousand steps for proving one program assertion, when it veried the control software of a telecommunication network consisting of about 10,000 lines of source code. This required proofs for many thousand assertions (and some of these proofs were then a little less than 10,000 steps long). The early verication of a complex hardware component such as the microprocessor FM8501 that was veried with the BoyerMoore System required about 150 pages of formulas of specications and lemmata in the Boyer Moore Logic and was carried out in about thirteen man-months. The complexity of today's verication task surpasses all of this by far. Dagstuhl Seminar Proceedings 05431 Deduction and Applications http://drops.dagstuhl.de/opus/volltexte/2006/561 2 J. Siekmann problem that they are usually written in the 'wrong' order. A human reader might say: 'they do not have a logical structure'; which is to say that of course they follow a logical pattern (as they are correctly generated by a machine), but, given the convention of the respective eld and the way the trained math- ematician in this eld is used to communicate, they are somewhat strange and ill structured. And nally, there is the problem that proofs are purely formal and recorded in a predicate logic that is very far from the usual presentation that relies on a mixture of natural language arguments interspersed with some formalism. The proof is far from such a presentation in the sense that even if all predicate, function and constant symbols were replaced by their natural language counter- part and even if all the logical formalisms were replaced by its natural language equivalent (such as 0∧0 and ')' by implies and so forth) the resulting proof in natural language would still be a far cry from what we consider natural, even it it were ingeniously augmented by the usual mathematical phraseology such as 'thus follows', 'em as can easily be seen' or 'now we have the following cases' followed by 'quod erat demonstrandum'. Is there a typical mathematical textbook with a universally accepted way of proving theorems? Style, notation and level of abstraction in the presentation of a mathematical argument have changed considerably over the centuries. They are still changing today. Moreover every mathematical discipline and even every subarea within this discipline has its own jargon and its battery of proof techniques and general methods, that make it distinct and immediately recognisable by an experienced mathematician of this eld. Apart from these considerations there is a more principle point to the problem of what constitutes a typical proof. Until the last century a mathematical proof was essentially a convincing argument a mathematician would use to persuade his colleagues in believing his theorem. While there was considerable controversy on the form of such an argument, dating back at least to Euclid's Elements, it essential means of presentation was that of natural language prose albeit highly stylised and augmented with some formal notation. Now what constitutes a 'convincing argument' is open to debate and mathematical truth was and in spite of all logicism still is established through a social process of conjectures and refutations carried out anew by each generation. This point of view was challenged and appeared to be irrevocably superseded by the turn of the century, when the foundational studies in mathematical logic and the slow but sure recognition of the importance of Frege's Begrisschrift marked the beginning of logicism. Hilbert's program with its success in the twenties and thirties seemed to render the whole of mathematics into a formal and essentially mechanical en- terprise: a proof is a formal object itself subject to mathematical rigour and analysis that proceeds from the hypothesis (the axioms) through a sequence of well understood and simple formal operations based on general accepted infer- ence rules, to its conclusion, the actual theorem. Although there is considerable Proof Presentation 3 variation within this point of view, which is the subject of proof theory (some of its standard material is covered in the rst part of this book) the essence of this holds universally true for all logical calculi: axioms and theorems are purely syn- tactical objects and the intermediate steps are based on formal rules of inference that could in principle be carried out mechanically on a suitable machine. Thus the touching seventeenth century prophecy of Gottfried Wilhelm Leib- niz that, when the new language is perfected, men of good will desiring to settle a controversy on any subject whatsoever will take their pens in their hands and say calculemus let us calculate, and carry out the argument in a purely formal manner within the calculus ratiocanator this vision had in principle become true for the mathematical disciplines by the middle of our century and logicism began to spread into other disciplines as well, as hallmarked inter alia by the Vienna Circle. As Martin Davis notes in his Prehistory and Early History of Automated Deduction, this view was not entirely unchallenged: Henry Poincare realised perfectly well that if the claim of the logicians were to be taken seriously, the possibility of mechanising human reason would be very real. But the very absur- dity of such a possibility which threatened everything creative and beautiful in mathematical thought, showed in fact the logicians claims need not be taken seri- ously. Poincare expressed his argument by reductio ad absurdum picturesquely as follows: Thus it will be readily understood that in order to demonstrate a theorem, it is not necessary or even useful to know what it means. We might replace geometry by the reasoning piano imagined by Stanley Jevons; or, if we prefer, we might imagine a machine where we should put in axioms at one end and take out theorems at the other, like the legendary machine in Chicago where pigs go in alive and come out transformed into hams and sausages. It is no more necessary for the mathematician than it is for these machines to know what he is doing. Be this as it may modern textbooks in many mathematical elds now reect the logical point of view: set theory, model theory, some books on algebra and many textbooks on logic itself are typical examples of this purely formal approach in the sense that their proofs are carried out entirely by syntactical operations on formulae often with very little natural language explanation in between. From Whitehead and Russell's Principia Mathematica to the Bourbaki group of mathematicians , there have been many attempts to reconstruct the whole of mathematics on the basis of a few basic principles, from which follows all that is known in mathematics by a few syntactical operations, i.e. logical inferences. The rst mechanical calculating devices for the four numerical operations, ad- dition, multiplication and their inverses, constructed by W. Schicard, B. Pascal and G. W. Leibniz in 1623, 1642 and 1671 respectively relied on two fundamental developments: the arithmetical calculus had developed from an art (for example calculating the payrolls for the Roman legionnaires) to a purely mechanical ap- plication of rules in the algorithms for the four basic arithmetic operations.